Two recent p-adic approaches towards the (effective) Mordell conjecture

Jennifer S. Balakrishnan; Alex J. Best; Francesca Bianchi; Brian Lawrence; J. Steffen Müller; Nicholas Triantafillou; Jan Vonk

doi:10.1007/978-3-030-65203-6_2

Two recent p-adic approaches towards the (effective) Mordell conjecture

Jennifer S. Balakrishnan^*, Alex J. Best, Francesca Bianchi, Brian Lawrence, J. Steffen Müller, Nicholas Triantafillou, Jan Vonk

^*Corresponding author for this work

Algebra

Research output: Chapter in Book/Report/Conference proceeding › Chapter › Academic › peer-review

11 Citations (Scopus)

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Abstract

We give an introductory account of two recent approaches towards an effective proof of the Mordell conjecture, due to Lawrence--Venkatesh and Kim. The latter method, which is usually called the method of Chabauty--Kim or non-abelian Chabauty in the literature, has the advantage that in some cases it has been turned into an effective method to determine the set of rational points on a curve, and we illustrate this by presenting three new examples of modular curves where this set can be determined.

Original language	English
Title of host publication	Regulators IV
Subtitle of host publication	An international conference on arithmetic L-functions and differential geometric methods
Editors	A Chambert-Loir, J-H. Lu, M. Ruzhansky, Y. Tschinkel
Place of Publication	Birkhäuser
Publisher	Springer
Pages	31-74
Number of pages	44
ISBN (Electronic)	978-3-030-65203-6
ISBN (Print)	978-3-030-65202-9
DOIs	https://doi.org/10.1007/978-3-030-65203-6_2
Publication status	Published - 2021

Publication series

Name	Progress in Mathematics
Publisher	Springer
Volume	338
ISSN (Print)	0743-1643

Keywords

math.NT
math.AG

Access to Document

10.1007/978-3-030-65203-6_2

Arithmetic L-Functions and Differential Geometric Methods. Progress in MathematicsFinal publisher's version, 6.41 MBLicence: Taverne

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Balakrishnan, J. S., Best, A. J., Bianchi, F., Lawrence, B., Müller, J. S., Triantafillou, N., & Vonk, J. (2021). Two recent p-adic approaches towards the (effective) Mordell conjecture. In A. Chambert-Loir, J.-H. Lu, M. Ruzhansky, & Y. Tschinkel (Eds.), Regulators IV: An international conference on arithmetic L-functions and differential geometric methods (pp. 31-74). (Progress in Mathematics; Vol. 338). Springer. https://doi.org/10.1007/978-3-030-65203-6_2

Balakrishnan, Jennifer S. ; Best, Alex J. ; Bianchi, Francesca et al. / Two recent p-adic approaches towards the (effective) Mordell conjecture. Regulators IV: An international conference on arithmetic L-functions and differential geometric methods. editor / A Chambert-Loir ; J-H. Lu ; M. Ruzhansky ; Y. Tschinkel. Birkhäuser : Springer, 2021. pp. 31-74 (Progress in Mathematics).

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abstract = " We give an introductory account of two recent approaches towards an effective proof of the Mordell conjecture, due to Lawrence--Venkatesh and Kim. The latter method, which is usually called the method of Chabauty--Kim or non-abelian Chabauty in the literature, has the advantage that in some cases it has been turned into an effective method to determine the set of rational points on a curve, and we illustrate this by presenting three new examples of modular curves where this set can be determined. ",

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Balakrishnan, JS, Best, AJ, Bianchi, F, Lawrence, B, Müller, JS, Triantafillou, N & Vonk, J 2021, Two recent p-adic approaches towards the (effective) Mordell conjecture. in A Chambert-Loir, J-H Lu, M Ruzhansky & Y Tschinkel (eds), Regulators IV: An international conference on arithmetic L-functions and differential geometric methods. Progress in Mathematics, vol. 338, Springer, Birkhäuser, pp. 31-74. https://doi.org/10.1007/978-3-030-65203-6_2

Two recent p-adic approaches towards the (effective) Mordell conjecture. / Balakrishnan, Jennifer S.; Best, Alex J.; Bianchi, Francesca et al.
Regulators IV: An international conference on arithmetic L-functions and differential geometric methods. ed. / A Chambert-Loir; J-H. Lu; M. Ruzhansky; Y. Tschinkel. Birkhäuser: Springer, 2021. p. 31-74 (Progress in Mathematics; Vol. 338).

Research output: Chapter in Book/Report/Conference proceeding › Chapter › Academic › peer-review

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T1 - Two recent p-adic approaches towards the (effective) Mordell conjecture

AU - Balakrishnan, Jennifer S.

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AU - Bianchi, Francesca

AU - Lawrence, Brian

AU - Müller, J. Steffen

AU - Triantafillou, Nicholas

AU - Vonk, Jan

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N2 - We give an introductory account of two recent approaches towards an effective proof of the Mordell conjecture, due to Lawrence--Venkatesh and Kim. The latter method, which is usually called the method of Chabauty--Kim or non-abelian Chabauty in the literature, has the advantage that in some cases it has been turned into an effective method to determine the set of rational points on a curve, and we illustrate this by presenting three new examples of modular curves where this set can be determined.

AB - We give an introductory account of two recent approaches towards an effective proof of the Mordell conjecture, due to Lawrence--Venkatesh and Kim. The latter method, which is usually called the method of Chabauty--Kim or non-abelian Chabauty in the literature, has the advantage that in some cases it has been turned into an effective method to determine the set of rational points on a curve, and we illustrate this by presenting three new examples of modular curves where this set can be determined.

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Balakrishnan JS, Best AJ, Bianchi F, Lawrence B, Müller JS, Triantafillou N et al. Two recent p-adic approaches towards the (effective) Mordell conjecture. In Chambert-Loir A, Lu JH, Ruzhansky M, Tschinkel Y, editors, Regulators IV: An international conference on arithmetic L-functions and differential geometric methods. Birkhäuser: Springer. 2021. p. 31-74. (Progress in Mathematics). Epub 2021 May 18. doi: 10.1007/978-3-030-65203-6_2

Two recent p-adic approaches towards the (effective) Mordell conjecture

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